Rigidity and functional properties of $\mathrm{BD}_{dev}(Ω)$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909627011039232 |
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| author | Caroccia, Marco Van Goethem, Nicolas |
| author_facet | Caroccia, Marco Van Goethem, Nicolas |
| contents | We provide a structural analysis of the space of functions of bounded deviatoric deformation, $\mathrm{BD}_{dev}$, which arises in models of plasticity and fluid mechanics. The main result is the identification of the annihilator and a rigidity theorem for $\mathrm{BD}_{dev}$-maps with constant polar vector in the wave cone characterizing the structure of singularities for such maps. This result, together with an explicit kernel projection operator, enables an iterative blow-up procedure for relaxation and homogenization problems, allowing for integrands with explicit dependence on $u$ as well as $\mathcal{E}_d u$. Our approach overcomes several difficulties as compared to the $\mathrm{BD}$ case, in particular due to the lack of invariance of $\mathcal{E}_d$ under orthogonalization of the polar directions. Applications to integral representation and Material science are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_23348 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidity and functional properties of $\mathrm{BD}_{dev}(Ω)$ Caroccia, Marco Van Goethem, Nicolas Analysis of PDEs Classical Analysis and ODEs We provide a structural analysis of the space of functions of bounded deviatoric deformation, $\mathrm{BD}_{dev}$, which arises in models of plasticity and fluid mechanics. The main result is the identification of the annihilator and a rigidity theorem for $\mathrm{BD}_{dev}$-maps with constant polar vector in the wave cone characterizing the structure of singularities for such maps. This result, together with an explicit kernel projection operator, enables an iterative blow-up procedure for relaxation and homogenization problems, allowing for integrands with explicit dependence on $u$ as well as $\mathcal{E}_d u$. Our approach overcomes several difficulties as compared to the $\mathrm{BD}$ case, in particular due to the lack of invariance of $\mathcal{E}_d$ under orthogonalization of the polar directions. Applications to integral representation and Material science are discussed. |
| title | Rigidity and functional properties of $\mathrm{BD}_{dev}(Ω)$ |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2505.23348 |